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Questions tagged [legendre-polynomials]

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3
votes
2answers
254 views

Legendre equation: An interpretation [closed]

I am a student of physics and, especially in quantum mechanics, we are presented with the Legendre equation: \begin{eqnarray} (1-x^2)y''-2xy'+l(l+1)y=0. \end{eqnarray} Doing some calculations, we ...
0
votes
0answers
37 views

A two argument sepcial function related to Legendre polynomial and Meixner polynomial

This problem raised when I was trying to evaluate a complicated integral. A polynomial with 2 arguments emerged and I could not recognize it. Let's call it $F_n(k,x)$, what I know is that $F_n(0,x)=...
2
votes
1answer
155 views

Integration of hypergeometric product for legendre polynomials

I'm looking for a general solution to the integral: $\int_{0}^1 {_2}F_1(m-n,m+n+1,m+1;z){_2}F_1(m-k+1,m+k+2,m+2;z) dz$ where $m,n,k\in \mathbb{N}$ and $m\leqslant n$ and $m+1 \leqslant k$. To give ...
2
votes
0answers
34 views

Rate of convergence of generalized polynomial chaos

Let $\eta=g(\xi_1,\ldots,\xi_M)$ be a random variable expressed as a function of the random vector $\xi=(\xi_1,\ldots,\xi_M)$. Assume that $\xi_1,\ldots,\xi_M$ are absolutely continuous and ...
3
votes
0answers
78 views

Identity for the product of two different associated Legendre polynomials

In the answer to Clausen’s identity for associated Legendre polynomials the following result was indicated: $$ \small{\left(P_n^m(\cos\theta)\right)^2=(\sin{\theta})^{2m}\frac{(m+n)!}{(n-m)!}\sum_{k=0}...
2
votes
1answer
287 views

Clausen’s identity for associated Legendre polynomials

Clausen’s identity for Legendre polynomials has the form (see, for example, A generating function of the squares of Legendre polynomials, by Wadim Zudilin: https://arxiv.org/abs/1210.2493) $$P_n(\cos{\...
2
votes
1answer
228 views

Bounds on Legendre polynomials on the complex plane

Let $P_n$ be the Legendre polynomial of degree $n$. Could you suggest me some references to bound the polynomials on the complex plane (near the real line in particular) ? More specifically, I need to ...
5
votes
2answers
576 views

Relation between Legendre and Chebyshev polynomials

Where I could find relationships between Legendre and Chebyshev polynomials? For example I found with maple $$ P_n(\cos\theta)=\sum_{k=0}^n(-1)^{n+k}\frac{2-\delta_{k0}}{4^n} \binom{n-k}{\frac{n-k}{2}...
2
votes
0answers
191 views

Formula of Laplace for the asymptotic expansion of Legendre polynomials

According to the so-called formula of Laplace (see, e.g., [1, Theorem 8.21.2]), Legendre polynomials admit the following asymptotic expansion: $$ P_n(\cos\theta) = \sqrt{2} (\pi n\sin\theta)^{-\frac12}...
1
vote
0answers
212 views

Legendre functions as Hypergeometric functions

Is there some approximation or regularization that goes in tacitly in the following equality: $Q_{\lambda }(z) = \frac{\sqrt{\pi } \Gamma (\lambda +1)}{2^{\lambda +1} \Gamma \left(\lambda +\frac{3}{2}...
2
votes
0answers
39 views

Legendre expansion of $r(x) = f(x)/g(x)$ using a finite number of samples from $f(x)$ and $g(x)$

I have two finite sets of events $\{x_1, ..., x_N\}$ and $\{y_1, ..., y_N\}$ that are sampled from the PDFs $f(x)$ and $g(x)$, respectively, where $x \in [-1,+1]$. I want to estimate the Legendre ...
1
vote
0answers
101 views

Expansion of prolate spheroidal harmonics

For two coordinate frames $O'$ and $O''$ both offset along the $z$-axis by $\pm R$ respectively, with corresponding offset spherical coordinates $r'$, $\theta'$, $r''$ and $\theta''$, and with prolate ...
5
votes
2answers
350 views

Recurrence of Legendre polynomial roots/ quadrature points

Consider Legendre polynomials $p_n (x)$ on $[-1,1]$. For each $n \in \mathbb{N}$ we denote the zeros of $p_n (x)$ by $\left( x_j ^{(n)} \right) _{j=1} ^n$. We know that these roots are distinct, and ...
0
votes
1answer
523 views

Integrals involving associated legendre polynomials

Do the following integrals have a closed-form solution for any integer value of $m,l,k$ and $n$? $\int^{\pi}_{0} P^{m}_{l}\left(\cos\theta\right)P^{n}_{k}\left(\cos\theta\right)\cot\theta d\theta$ $\...
7
votes
2answers
2k views

Legendre Polynomial Integral

How can I evaluate $$ \int_{-1}^1 P_n(x)P_l(x)x^k dx $$ when $k$ is even? Or what might be a source where I could find integrals like this?
6
votes
2answers
269 views

Symmetric matrix formula for Gauss-Legendre quadrature

While searching the web, I came across the following algorithm for the Gauss-Legendre quadrature. I wasn't able to find a reference or a proof of my own as to why it works. I'll present it, and the ...
3
votes
1answer
380 views

Reference for the exponential decay of Legendre coefficients

In Short: I look for a reference to the proof that the spectral coefficients in the Legendre (or Jacobi) expansion are of exponential decay rate. Longer: If $p_n$ is the $n$-th Legendre polynomial, ...
0
votes
0answers
339 views

Orthogonality of Associated Legendre polynomials for upper indices

Can anyone give me a hint how to prove the following property of the associated Legendre polynomials $$\int_{-1}^{+1}P_{n}^{m}(x)P_{n}^{k}(x)\frac{dx}{1-x^2}=\frac{(n+m)!}{m(n-m)!}\delta_{mk}$$ for $...
1
vote
0answers
217 views

Legendre equation with homogeneous boundary condition

I'm looking for solutions to the Legendre differential equation $$ \frac{d}{dx}\left((1-x^2)\frac{d}{dx}f(x)\right)+(\alpha-1)\alpha f(x)=0 $$ with boundary conditions $f'(0)=0$ and $f(1)=0$, but ...
15
votes
4answers
645 views

Maximum of the Vandermonde determinant / minimum of the logarithmic energy

The problem is to find the asymptotics (as $n\to\infty$) of the maximum (say $M_n$) of the Vandermonde determinant $$V_n:=\prod_{0\le i<j\le n-1}(a_j-a_i) $$ over all $a_0,\dots,a_{n-1}$ such ...
11
votes
2answers
652 views

Computing Gauss Legendre Quadrature for Large N

I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscissas and weights $\{ x_j, w^j \} _{j=1}^N$ for large $N\in\mathbb{N}$. My question is how to do it,...
0
votes
1answer
121 views

Equality cannot hold unless $x \in \{-1,1\}$ and/or Wronskian is not zero [closed]

By playing around with assoc. Legendre polynomials, I arrived at $$((l+1)+m) (P_l^m(x))^2+((l+1)-m)(P_{l+1}^m(x))^2 = 2(l+1)x P_l^m(x)P_{l+1}^m(x).$$ Now, I want to show that we don't have equality ...
0
votes
1answer
312 views

Integral Transform with associated Legendre Function of second kind as kernel

In my research the following equation appeared: $$\frac{1}{4\pi}\int_{0}^{1}\frac{t^{s-1}(1-t)^{s-1}}{(\rho-t)^s}dt=\int_0^{\infty} f(a) Q^{i\sqrt{a}}_{s-1}(2\rho-1) da,$$ where $\rho,s>1$, $Q^{\...
5
votes
0answers
249 views

Legendre polynomials and formal groups

Let $P_n(x)$ be Legendre polynomials: $$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\limits_{n=0}^{\infty}P_n(x)t^n.$$ Usual arguments from the theory of formal groups allow to prove that for any $n$ $$P_n(x)=Q_n(...
1
vote
0answers
105 views

Fourier-Legendre coefficients and Sobolev regularity

Let $f$ be an $L^1$-function supported in $[-b, b]$, where $0 < b \le b_0 < 1$. Let $Q_n$ be the normalised Legendre polynomials and let $a_n = \int f(x) Q_n(x) dx$ be the Fourier-Legendre ...